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SQUARE: Structured Quantum Representation Adapters as Compact Quadratic Feature Maps for Frozen Language Models

Emily Jimin Roh, Hyojun Ahn, Hoyeong Lee, Soohyun Park, Sung Whan Yoon, Vaneet Aggarwal, Joongheon Kim

arXiv:2609.37134v1quant-phcs.AI

TL;DR

Frozen language-model bottlenecks leave a compact downstream module to represent dependencies among their coordinates, while linear adapters do not explicitly expose pairwise products. SQUARE applies a trainable circuit to an amplitude-encoded bottleneck and measures normalized quadratic features; in the reported controlled interaction comparison, it outperforms the listed alternatives. The evidence remains limited to controlled interaction settings rather than establishing broad gains on natural-label NLP tasks.

  • Problem

    Frozen-LM bottlenecks need compact downstream maps that expose target-relevant coordinate dependencies, which linear and low-rank adapters do not explicitly represent as pairwise products.

    Source: §1 · p. 1, §3 · p. 4

  • Method

    SQUARE amplitude-encodes a frozen-LM bottleneck, applies a trainable unitary, and measures features whose basis probabilities are normalized quadratic forms.

    Source: §1 · p. 2, §3.2 · pp. 4–5

  • Results

    0.7565 average test accuracy for SQUARE exceeds 0.7355 for an affine normalized-quadratic score, 0.7271 for parameter-matched Givens-12, 0.6817 for an MLP, and 0.6155 for a frozen circuit.

    Source: §1 · p. 3

  • Takeaways & Limitations

    SQUARE is a structured parameterization within the classical normalized-quadratic feature family, and its learned map can be evaluated in batched PyTorch without quantum hardware.

    Source: §1 · p. 2, §5 · p. 9

    The evidence is limited to low-dimensional controlled interaction settings and does not establish broad gains on natural-label NLP tasks.

    Source: §4.2 · p. 9, §5 · p. 10

Abstract

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Frozen language models (LMs) are increasingly used as fixed feature extractors for downstream reranking, scoring, and preference modeling, raising a practical question: how should a compact module represent interactions among features in a fixed low-dimensional bottleneck? Common linear and low-rank adapters remain linear at the adaptation module itself, whereas explicit second-order alternatives introduce pairwise interactions through direct parameterization or predefined factorizations. We propose SQUARE, a Structured QUAntum REpresentation adapter that amplitude-encodes the bottleneck vector, applies a parameterized quantum circuit, and measures the resulting state. We show that each basis-probability feature is exactly a normalized quadratic form in the bottleneck coordinates, while the additional Pauli-$Z$ readouts are signed linear combinations of these probabilities. The measured map can therefore parameterize interactions over $O(d^2)$ coordinate pairs through a small set of shared circuit parameters, where $d$ is the bottleneck dimension. It provides a structured parameterization within, rather than beyond, the classical normalized-quadratic feature class. In a disjoint same-pipeline evaluation over eight GLUE-derived controlled interaction tasks and five shared seeds, SQUARE achieves an average test accuracy of $0.7565$, compared with $0.7355$ for an affine normalized-quadratic predictor, $0.7271$ for the evaluated parameter-matched Givens mixing model, $0.6817$ for an MLP, and $0.6155$ for a frozen-circuit control. Under reduced supervision, it also shows consistent gains over the strongest evaluated classical comparator, with the same qualitative pattern across multiple frozen LM backbones. All circuit experiments use simulation, while the learned feature map can be evaluated exactly in batched PyTorch without quantum hardware.

1 INTRODUCTION

The introduction frames adaptation over frozen LM bottlenecks as a question of representing feature interactions, then presents SQUARE as a shared-circuit parameterization of normalized quadratic features. Controlled evaluations favor SQUARE among the reported alternatives, while its learned map can be deployed through classical PyTorch.

1 INTRODUCTION

  • Frozen LM bottlenecks require a compact downstream map that makes target-relevant dependencies among coordinates accessible to a lightweight task head.Linear, bias-only, and low-rank adaptation maps remain first order rather than explicitly exposing pairwise products.Source: §1 · p. 1
  • SQUARE’s circuit-defined feature map is compiled into batched PyTorch for deployment, without requiring quantum hardware.The circuit is evaluated with simulation; the paper frames its value as a structured learning principle, not a uniquely quantum function class.Source: §1 · pp. 2–3
  • SQUARE amplitude-encodes the bottleneck, applies a trainable unitary, and measures it; each basis-probability feature is a normalized quadratic form.One shared unitary jointly generates the probability feature bank, whose coefficient matrices are low-rank PSD forms that collectively resolve the identity.Source: §1 · p. 2
  • The evaluation uses GLUE-derived controlled interaction labels and also reports original-label GLUE, reranking, circuit-ablation, and simulated-noise diagnostics.These protocols distinguish controlled interaction tests from broader task-oriented diagnostics.Source: §1 · pp. 2–3
  • 0.7565 average test accuracy for SQUARE exceeds 0.7355 for an affine normalized-quadratic score, 0.7271 for parameter-matched Givens-12, 0.6817 for an MLP, and 0.6155 for a frozen circuit.The primary comparison is disjoint and same-pipeline; the evaluated controlled protocols also include reduced supervision and multiple frozen backbones.Source: §1 · p. 3

2 RELATED WORK

Related work spans parameter-efficient language-model adaptation, classical interaction models over fixed representations, and hybrid quantum methods. SQUARE instead learns a jointly constrained normalized probability feature bank over a compact projected subspace.

2 RELATED WORK

  • Many parameter-efficient methods adapt Transformer weights, whereas SQUARE keeps the backbone fixed and learns downstream of a frozen bottleneck.Transformer-weight adaptation methods are broad references rather than the mechanism-matched comparison set; evaluated LoRA- and adapter-style modules operate at the same bottleneck stage.Source: §2 · p. 3
  • Classical polynomial, bilinear, factorization, and kernel models already provide second-order interaction methods over fixed embeddings.SQUARE’s narrower distinction is a shallow complex-unitary circuit that jointly generates a normalized probability feature bank; quadraticity itself is not its novelty.Source: §2 · p. 3
  • Unlike QAA’s full-hidden-state encoding or QPA’s PEFT-weight generation, SQUARE operates on a compact projected subspace and exposes measured interaction features.The related-work passage also distinguishes QuIC as adapting Transformer weights with quantum-inspired orthogonal adapters.Source: §2 · pp. 3–4

3 SQUARE

SQUARE learns a compact predictor over a fixed frozen-LM bottleneck by giving its task head measured features that explicitly represent cross-coordinate interactions. Its circuit-induced map is characterized as a constrained quadratic feature family and is trained and deployed through classical implementations.

3 SQUARE

  • SQUARE keeps the language model and bottleneck projection fixed, learning only the adaptation module and task-head parameters.The measured features provide explicit second-order interactions before the lightweight head, which can output logits, scalar scores, or ranking scores.Source: §3 · p. 4, §3.1 · p. 4

3.1 PROBLEM FORMULATION

  • Linear, bias-only, and low-rank bottleneck transformations do not explicitly form pairwise products, while SQUARE supplies measured cross-amplitude features for that structure.The formulation separates coordinate-wise evidence from pairwise and higher-order relations.Source: §3.1 · p. 4

3.2 SQUARE: MEASURED QUANTUM FEATURE MAP

  • Amplitude encoding normalizes the bottleneck, after which a trainable rotation–entanglement–rotation circuit produces probability and local Pauli-Z readouts.Each probability feature is exactly quadratic with reconstruction R2 = 1.0000 and maximum residual below 5×10^-7; its coefficient matrix is PSD and rank at most two, and the probability matrices sum to identity. Amplitude encoding preserves angular geometry of nonzero bottleneck directions, not norm magnitude.Source: §3.2 · pp. 4–6

3.3 TRAINING AND PARAMETERIZATION

  • Training jointly optimizes circuit angles and the task head through simulator backpropagation; the learned gate product is evaluated in batched PyTorch without per-sample simulator calls.The two inference paths agree to 1.118 × 10^-7; the circuit has 3Ln trainable quantum parameters at depth L over n qubits.Source: §3.3 · p. 6

4 EXPERIMENTS

The experiments test whether SQUARE’s circuit-induced coefficient structure helps recover controlled interactions over fixed language-model bottlenecks. In the held-out eight-task comparison, SQUARE leads the evaluated same-pipeline controls, while the paper limits that conclusion to controlled interaction settings.

4.1 DIAGNOSTIC NONLINEAR DECISION BOUNDARY ANALYSIS

  • On the MNLI-derived PCA boundary task, SQUARE achieves the best Accuracy, F1, and AUC, including AUC 0.8259 versus 0.7217 for MLP and 0.6734 for Prefix.This visualization pipeline includes a learned nonlinear projection, so it does not establish the main coefficient-structure claim.Source: Table 1 · p. 7, §4.1 · p. 8
  • Across checkerboard and radial-ring tasks, SQUARE reaches average test AUC 0.992, compared with 0.980 for classical Fourier and 0.975 for explicit polynomial maps.The checkerboard target lies outside the degree-two family, so this diagnostic tests composition by the nonlinear head rather than direct membership in SQUARE’s feature class.Source: §4.1 · p. 8, Table 2 · p. 8

4 EXPERIMENTS

  • The protocol applies one prespecified interaction-rich rule to frozen representations from eight GLUE datasets, using disjoint splits and five shared seeds.Representations, bottleneck features, labels, and evaluation conditions are shared across methods, with validation-only model selection.Source: §4 · p. 7, §4.2 · p. 8

4.2 GLUE-DERIVED CONTROLLED INTERACTION CLASSIFICATION: CIRCUIT-INDUCED QUADRATIC BIAS VS. CLASSICAL PARAMETERIZATIONS

  • SQUARE leads the eight-task held-out comparison, exceeding Givens-12 by 0.029 average accuracy and the affine normalized-quadratic score by 0.021.The Givens-12 comparison matches parameter count, squared features, and head; the normalized-quadratic comparison also differs in head composition.Source: Table 3 · p. 8, §4.2 · p. 9
  • The held-out ordering supports a useful inductive bias from the jointly constrained unitary-measurement parameterization within the normalized-quadratic family, not universal superiority or quantum computational advantage.The coefficient-sharing interpretation is limited to implemented controls and the controlled target family.Source: §4.2 · p. 9
  • The qualitative SQUARE–MLP ordering persists across four additional frozen backbones, while original-label GLUE results support parameter compactness rather than an accuracy advantage.The controlled-label gains should not be generalized to natural-label classification; the additional-backbone experiments retrain adapters for each representation.Source: §4.2 · p. 9

5 CONCLUSION

SQUARE is presented as a structured parameterization of normalized quadratic features over frozen language-model bottlenecks, with controlled experiments supporting gains over the evaluated alternatives. The conclusion confines the evidence to low-dimensional controlled interactions and calls for broader validation.

5 CONCLUSION

  • SQUARE parameterizes normalized quadratic features over frozen language-model bottlenecks, with a shared unitary jointly constraining its probability features.The conclusion describes it as a structured parameterization within the classical normalized-quadratic family, not a uniquely quantum function class or source of computational advantage.Source: §5 · p. 9
  • The evidence is limited to low-dimensional controlled interaction settings and does not establish broad gains on natural-label NLP tasks.The conclusion identifies natural supervision, higher-dimensional bottlenecks, coordinate-permuted protocols, and additional structured parameterizations as directions for validation.Source: §5 · p. 10

AI USE STATEMENT

  • The authors report using generative AI tools for manuscript language refinement, proofreading, and presentation clarity, while retaining responsibility for the content.They state that they independently reviewed and verified the scientific claims, derivations, designs, implementations, results, and references.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The paper specifies that its experimental pipeline can be reproduced without quantum hardware, and points to appendices for protocols and the direct PyTorch realization.The stated materials include data construction, splits, label rules, baseline configurations, model selection, metrics, and hyperparameters.Source: REPRODUCIBILITY STATEMENT · p. 10

ETHICS STATEMENT

  • Because SQUARE uses frozen language-model representations, it may inherit backbone biases, spurious correlations, or other limitations.The ethics statement recommends task-specific validation, fairness and reliability checks, privacy safeguards, and human oversight where applicable.Source: ETHICS STATEMENT · p. 10

A BACKGROUND ON QUANTUM MACHINE LEARNING

  • The background section presents SQUARE as a hybrid pipeline that encodes a classical representation, transforms the resulting state with a parameterized circuit, and converts it to classical features through measurement.It reviews qubits, Hilbert-space representations, amplitude encoding, trainable circuits, and measurement readouts in pipeline order.Source: A BACKGROUND ON QUANTUM MACHINE LEARNING · p. 15

A.1 QUBITS AND HILBERT-SPACE REPRESENTATIONS

  • A qubit is represented by a normalized vector in a two-dimensional complex Hilbert space, and computational-basis measurement probabilities are squared amplitude magnitudes.For the single-qubit basis states, the stated probabilities are |α|^2 for |0⟩ and |β|^2 for |1⟩.Source: §A.1 · p. 15

A.2 PARAMETERIZED QUANTUM CIRCUITS

  • SQUARE applies trainable RY and RZ rotations, ring CNOT entanglement, and a final RY block before measurement.The illustrated example uses four qubits and one layer; the stated ring uses n CNOT gates per layer to connect all qubits.Source: §A.2 · p. 16, Figure 3 · p. 16

A.4 STRUCTURAL CHARACTERIZATION AND SCALE INVARIANCE OF THE MEASURED MAP

SQUARE’s probability readouts form a jointly constrained family of normalized quadratic features, with scale-invariance and circuit structure delimiting what the map can represent. The reported comparisons show competitive interaction recovery under compact budgets and preserve frozen-bottleneck utility on original GLUE labels.

A.4 STRUCTURAL CHARACTERIZATION AND SCALE INVARIANCE OF THE MEASURED MAP

  • Each probability feature is a PSD, rank-at-most-two normalized quadratic form, and the collection resolves identity while sharing one unitary.At n=4, observed local Jacobian ranks are 12, 20, and 28 for depths 1, 2, and 3; these are not global-dimension claims.Source: §A.4 · pp. 18–19
  • Because the measured features are invariant to nonzero input scaling, a predictor using them cannot distinguish points on the same ray or exactly represent a norm-dependent label rule.The GLUE-derived controlled tasks and additional-backbone experiments use the pure path on a d=16 PCA bottleneck.Source: §A.4 · p. 18
  • On the disjoint held-out pure-path suite, training the circuit raises mean accuracy from 0.6155 to 0.7565 and exceeds the normalized-quadratic and Givens-12 controls.The paper interprets a random circuit as an anisotropic, coordinate-aligned quadratic feature map, while cautioning that the held-out comparison does not prove a finite-sample regularization effect.Source: §A.4 · p. 19

A.5 HYBRID QUANTUM-CLASSICAL TRAINING DETAILS

  • Training jointly optimizes the measured features and classical head through simulator backpropagation, then compiles the learned unitary into numerically equivalent batched PyTorch inference.The circuit amplitude-encodes the bottleneck, applies rotation–ring-entanglement–rotation layers, and returns basis probabilities plus Pauli-Z expectations.Source: §A.5 · p. 19, §A.6 · pp. 19–20

A.7 TRENDS IN RECENT QUANTUM-ENHANCED LEARNING STUDIES

  • The surveyed QML studies commonly use hybrid architectures, task-specific encodings, and simulators; SQUARE applies this pattern to frozen semantic bottleneck features without claiming hardware advantage.Its measured readouts are used for downstream prediction rather than quantum parameter generation or generic activation adaptation.Source: Table 6 · p. 21, §A.7 · p. 21

B.1 DECISION-BOUNDARY EVALUATION DETAILS

  • The boundary appendix reports per-geometry test AUC comparisons over six seeds and per-method Accuracy, F1, and AUC for the two-dimensional PCA task.The section distinguishes controlled interaction-label recovery from predictive utility on restored GLUE labels with the LM and bottleneck frozen.Source: §B.1 · p. 22, Table 7 · p. 22, §B.2 · p. 22

B.2 COMPACT-BUDGET AND ORIGINAL-LABEL GLUE COMPARISONS

  • SQUARE attains a 0.969 macro-average with 68 trainable parameters, while the factorization machine is within 0.001 using 71 parameters on the compact-budget comparison.SQUARE also gives the strongest observed results on RTE and MRPC; the authors attribute the comparison to interaction-coefficient structure rather than parameter count alone.Source: §B.2 · p. 22
  • On original labels, SQUARE matches the frozen-bottleneck MLP average of 0.686 using 49 rather than 5,041 trainable parameters, while encoder-adapted LoRA remains more accurate.The reported comparison supports parameter-efficient interaction adaptation, not an original-label accuracy advantage over the MLP.Source: §B.2 · p. 22

B.3 POST-TRAINING SIMULATED NOISE SENSITIVITY

Post-training simulated noise produces mild degradation across the tested channels, while a separate broad frozen-bottleneck suite places SQUARE among the strongest observed methods. Both findings are scoped to simulator-based evaluations.

B.3 POST-TRAINING SIMULATED NOISE SENSITIVITY

  • The noise experiment fixes a model trained on a noiseless simulator and perturbs circuit evaluation with four standard noise channels to assess inference-time sensitivity.It does not claim robustness on physical quantum hardware.Source: §B.3 · p. 23, Figure 4 · p. 23
  • The largest AUC drop is 0.0103 under bit-flip noise; depolarizing, amplitude-damping, and phase-flip yield maximum drops of 0.0071, 0.0079, and 0.0032.Noise is injected only during evaluation after clean training; the authors treat the results as a sensitivity diagnostic, not a calibrated hardware-noise benchmark.Source: §B.3 · p. 23, Figure 4 · p. 23

B.4 BROAD FROZEN-BOTTLENECK APPROXIMATOR SUITE

  • In the broad shared frozen-bottleneck comparison, SQUARE has the highest observed mean AUC but belongs to a strongest-performance group because leading methods’ uncertainties overlap.The evaluation uses shared data, representations, splits, labels, and metric, with validation-only selection.Source: §B.4 · p. 23

B.5 ADDITIONAL BACKBONE RESULTS

Across the evaluated frozen backbones, SQUARE leads on controlled interaction classification, and its measured features also improve controlled candidate reranking. Gains vary by task and backbone, and the results are specific to these controlled settings.

Learning Dynamics on Additional Backbones

  • Across OPT-350M and GPT-2, SQUARE reaches the strongest final validation accuracy with stable optimization, while MLP remains the strongest classical baseline.SQUARE without the PQC, QAA, and QPA remain below SQUARE; the figure reports validation accuracy averaged across tasks with task-wise variation shaded.Source: Figure 5 · p. 24, §B.5 · p. 25

Additional Frozen Backbones

  • 0.8294 vs. 0.7960 on OPT-350M and 0.8258 vs. 0.7847 on GPT-2: SQUARE has the best average accuracy on both backbones.The comparisons are against MLP on GLUE-derived controlled interaction classification.Source: §B.5 · p. 25, Table 12 · p. 25, Table 13 · p. 26
  • The controlled-interaction results span OPT, GPT-2, OpenLLaMA, and Mistral, but do not establish superiority on arbitrary downstream objectives or natural-label tasks.Task-wise gains vary across backbones because their bottleneck geometries preserve different task-relevant interaction structure.Source: Table 11 · p. 24, §B.5 · pp. 25–26

Scaling to Billion-Parameter Backbones

  • 0.7714 vs. 0.7052 on OpenLLaMA-3B and 0.7481 vs. 0.7012 on Mistral-7B: SQUARE’s macro-average accuracy exceeds MLP on both backbones.SQUARE uses 157 trainable parameters; the comparison is reported for the larger frozen-backbone study.Source: §B.5 · p. 26, Table 14 · p. 27, Table 15 · p. 27
  • SQUARE leads on seven of eight OpenLLaMA-3B tasks and six of eight Mistral-7B tasks, while MLP wins WNLI and STS-B/WNLI, respectively.Gain magnitudes vary with task and backbone representation geometry.Source: §B.5 · p. 26, Table 14 · p. 27, Table 15 · p. 27

Controlled Reranking on Larger Backbones

  • On OpenLLaMA-3B, SQUARE (Quantum) and SQUARE (Hybrid) each reach candidate AUC 0.745 versus 0.736 for MLP; Hybrid also has the highest Label Accuracy and F1.The hybrid values are 0.708 and 0.724, respectively, versus 0.707 and 0.701 for MLP.Source: §B.5 · p. 26
  • On Mistral-7B-v0.1, SQUARE (Hybrid) leads MLP in Label Accuracy (0.731 vs. 0.727), F1 (0.713 vs. 0.704), and candidate AUC (0.779 vs. 0.760).SQUARE (Quantum) reaches candidate AUC 0.778, while its Label Accuracy and F1 remain below MLP and Hybrid.Source: §B.5 · p. 26
  • Across both larger-backbone controlled reranking evaluations, both SQUARE variants improve candidate AUC over MLP, while Hybrid leads in Label Accuracy and F1.The comparison is parameter-matched and uses the MRPC-derived controlled reranking task.Source: §B.5 · p. 27

B.6 COMPUTE-RESOURCE BENCHMARK

The benchmark separates simulator training overhead from the cost of a classical realization of the learned map. The PyTorch implementation closely reproduces circuit features, but parameter compactness does not imply computational efficiency.

Adapter-Level Runtime Benchmark

  • 21.3373 ms per sample during training and 46.9 samples/s: SQUARE’s PennyLane simulator implementation has substantial overhead despite using 180 trainable parameters.BitFit, LoRA, Prefix, and MLP each remain below 0.07 ms per sample during training in the adapter-level benchmark.Source: §B.6 · pp. 27–28, Table 17 · p. 28

Classical Realization

  • At batch size 8 on an NVIDIA H200, PyTorch reproduces simulated features within 1.118 × 10^-7 and cuts forward latency from 97.598 ms to 4.720 ms (20.7×).The compiled module avoids simulator calls and quantum hardware at inference, but does not establish a speed advantage over specialized classical adapters.Source: §B.6 · p. 28

Matched-Protocol Training Benchmark

  • The matched training benchmark compares PennyLane backpropagation, native PyTorch statevectors, and an MLP under the same circuit, head, batch, precision, and CPU protocol.It measures forward, backward, and an Adam step; the shared host makes ratios more informative than absolute times.Source: §B.6 · p. 28, Table 18 · p. 29
  • On 64 random inputs, the two SQUARE paths agree to 4.8 × 10^-9 in features and 2.1 × 10^-8 in angle gradients, while native PyTorch remains slower than the MLP.The reported residual is attributed to the ε=10^-8 normalization constant.Source: §B.6 · pp. 28–29

B.7 GENERATION-ORIENTED RERANKING: FULL RESULTS

The Alpaca reranking evaluation finds approximate parity with compact classical scoring rather than a reliable scorer win. Its results support transfer to a reranking objective, but not improved human-perceived quality or a specifically quantum benefit.

Candidate-Level Results

  • For 120 instructions with eight candidates per pool, SQUARE’s scorer is numerically highest, but its gaps from MLP are not separable under instruction-paired bootstrap.The NonlinearScore difference is +0.012 (95% CI [−0.012, 0.036]); the HitRate difference is +0.033 ([−0.033, 0.100]).Source: §B.7 · p. 29, Table 19 · p. 29
  • Random selection reaches NonlinearScore 0.4991 and Oracle-ρ reaches 0.6890; the pure path reaches 0.5222 at the cost of lexical metrics.The evaluation selects each method’s top-ranked response from a shared candidate pool scored over the same frozen BERT bottleneck.Source: §B.7 · p. 29, Table 19 · p. 29

Interpretation

  • The results support transfer to reranking and approximate parity with compact classical scoring, but not improved human-perceived quality or a specifically quantum benefit.The candidate pool is fixed across methods, so the evaluation focuses on scorer differences rather than generator differences.Source: §B.7 · p. 29

B.8 LIMITATIONS AND FUTURE DIRECTIONS

The paper frames SQUARE’s evidence as controlled interaction recovery, with broader natural-supervision accuracy gains still unestablished. It also identifies compute and scaling limits, while emphasizing deployment through a classical PyTorch realization.

B.8 LIMITATIONS AND FUTURE DIRECTIONS

  • The eight GLUE-derived entries are eight frozen-representation distributions under one shared constructed rule, not eight independent semantic tasks.The authors say natural-label results currently support compactness rather than an accuracy gain.Source: §B.8 · p. 29
  • Simulation adds training overhead, and the native PyTorch realization remains slower than the matched MLP despite exact, simulator-free deployment.The authors explicitly caution that parameter efficiency should not be read as computational efficiency.Source: §B.8 · pp. 29–30
  • The central experiments use d=16 bottlenecks and four-qubit statevectors, so the additional-backbone study does not test dimension scaling.A dense classical realization may require O(B^2) storage and arithmetic despite the chosen template’s O(L log d) stored angle count and probability-readout width B < 2d.Source: §B.8 · p. 30

B.9 BROADER IMPACT

  • The intended workflow trains with a circuit parameterization, then deploys the learned map as an exact classical PyTorch module without updating the frozen backbone.The authors state that this is not a claim of near-term quantum hardware advantage.Source: §B.9 · p. 30
  • SQUARE can inherit or amplify biases and spurious correlations in frozen representations, requiring task-specific validation, fairness and robustness checks, privacy safeguards, and human oversight.The paper highlights sensitive domains including education, healthcare, hiring, and legal support.Source: §B.9 · p. 30

B.10 ADDITIONAL BOUNDARY CONTROLS: UNTRAINED-CIRCUIT, FINITE-SHOT, AND CLASSICAL COLLAPSE

Boundary controls show finite-shot readout sensitivity is small in the stated tutoring pipeline, while trained angles are not decisive there. In the disjoint eight-task evaluation, SQUARE leads the same-pipeline controls; the section also clarifies parameter counts and their interpretation.

B.10 ADDITIONAL BOUNDARY CONTROLS: UNTRAINED-CIRCUIT, FINITE-SHOT, AND CLASSICAL COLLAPSE

  • On checkerboard, depth-1 SQUARE reaches mean test AUC 0.849 ± 0.039, versus 0.609 ± 0.013 for RBF-SVM and 0.484–0.501 for the listed MLP, adapter, LoRA, and random-forest controls.This boundary pipeline includes a nonlinear projection and magnitude readout, so the result does not isolate an exactly quadratic predictor.Source: §B.10 · p. 30, Table 20 · p. 31
  • Across five boundary geometries, trained-minus-untrained AUC gaps stay within [−0.017, +0.022], unlike the larger trained-versus-frozen separation in the held-out GLUE-derived suite.The boundary observation is specific to the residual, learned-projection protocol and does not isolate a degree-two end-to-end predictor.Source: §B.10 · p. 30
  • At 100 shots, checkerboard depth-1 SQUARE changes from AUC 0.849 to 0.833; at 1,000 and 10,000 shots it returns to 0.849.Across other geometries, the 100-shot change is at most approximately 0.01; this is a sensitivity check, not evidence of exact quadraticity.Source: §B.10 · pp. 30–31, Table 21 · p. 31

B.11 TRAINABLE-PARAMETER ACCOUNTING AND HEAD ARCHITECTURES FOR SQUARE

  • The primary held-out configuration has 157 independently optimized parameters, while a separate 180-parameter training configuration folds exactly to a 148-scalar inference representation.The fold preserves the trained function but not identical optimization; the 148-scalar count is not the trained model size.Source: §B.11 · pp. 31–32, Table 22 · p. 32

B.12 DISJOINT HELD-OUT VALIDATION OF THE SAME-PIPELINE CONTROLS

  • SQUARE achieves the highest eight-task mean, exceeding the normalized-quadratic predictor, parameter-matched Givens-12 model, MLP, and frozen-circuit control.Taskwise interval summaries consistently favor SQUARE, but are descriptive rather than calibrated confidence intervals for the macro-average.Source: §B.12 · p. 32, Table 23 · p. 32

B.13 POST-ENTANGLEMENT CIRCUIT CONFIGURATION ANALYSIS

The circuit diagnostic favors the prespecified rotation–topology combination on three held-out tasks, but its scope remains limited to that diagnostic. The accompanying appendices specify the controlled datasets, label generators, reranking protocol, and evaluation metrics.

B.13 POST-ENTANGLEMENT CIRCUIT CONFIGURATION ANALYSIS

  • The prespecified RY RZ–ring configuration reaches mean accuracy 0.8462 across QNLI, MNLI, and RTE, exceeding the tested non-entangling, linear-CNOT, and all-to-all variants.The RY–ring variant trails both RY–none and RY–linear, so the result supports the jointly specified rotation–topology design rather than entanglement alone.Source: §B.13 · p. 33, Table 24 · p. 33

C.1 DATASET CONSTRUCTION AND SPLITS

  • Main experiments use frozen BERT bottleneck representations; controlled tasks generate labels from nonlinear interaction rules, while reranking methods share fixed candidate pools.The reranking comparison therefore evaluates scoring over the same candidates, not candidate generation.Source: §C.1 · p. 33, Table 25 · p. 33

C.2 CONTROLLED DECISION-BOUNDARY DATA

  • The two-dimensional boundary dataset uses 1,000 nominal training and 400 validation examples per seed, with near-threshold points removed separately from each split.Retained denominators vary by seed and geometry after excluding examples within 0.05 of the threshold.Source: §C.2 · pp. 33–34, Table 26 · p. 34

C.3 EXACT GENERATORS FOR THE DECISION-BOUNDARY GEOMETRIES

  • The boundary geometries are not degree-two polynomials or single bilinear thresholds, and the learned tanh projection can encode radius into direction for the radial-ring rule.A direct map is scale invariant and cannot represent that rule; this is a property of the direct map, not a mathematical necessity for the broader evaluated architecture.Source: §C.3 · p. 34

C.4 GLUE-DERIVED CONTROLLED INTERACTION DATA

  • The eight GLUE-derived tasks use one shared constructed interaction rule over frozen-representation distributions, not eight independent semantic targets.The rule includes sinusoidal and cosine product terms and is not exactly degree two; performance is distribution-conditional and does not show that targets belong to SQUARE’s hypothesis class.Source: §C.4 · pp. 34–35

C.5 OPERATIONAL DEFINITIONS FOR THE ALPACA RERANKING PROTOCOL

  • Reranking scores each instruction–candidate pair and selects the highest-scored candidate; NonlinearScore measures an author-specified interaction target, not externally judged response quality.The anchor candidate’s q+2.0 training target is unreachable under the sigmoid output and acts as a shared margin-inducing heuristic.Source: §C.5 · p. 35, §C.7 · p. 37

C.6 ALPACA CANDIDATE RERANKING DATA

  • The Alpaca reranking split contains 960 training, 120 validation, and 120 test queries, with the same eight candidates supplied to every method.The setup evaluates reranking quality rather than generation quality.Source: §C.6 · p. 36, Table 27 · p. 36

C.7 EVALUATION METRICS

  • Evaluation metrics vary by task; for circuit configuration analysis, AUC is primary because it assesses nonlinear separability without a fixed classification threshold.NonlinearScore instead assesses agreement with the specified interaction-dependent target over frozen coordinates, not external response quality.Source: §C.7 · pp. 36–37

C.8 BASELINE IMPLEMENTATION

The baseline suite compares adaptation mechanisms over a shared frozen representation space. It includes classical and quantum-enhanced references, interaction-specific classical models, and an internal SQUARE ablation.

C.8 BASELINE IMPLEMENTATION

  • The encoder and bottleneck projection remain frozen across methods, so optimization is limited to adapter-specific parameters and the task head.Hyperparameter selection uses the compact ranges reported for each baseline and SQUARE.Source: §C.8 · pp. 37–38, Table 28 · p. 39
  • BitFit, LoRA, AdaLoRA, and Prefix represent classical PEFT references, while a larger-budget MLP tests compact nonlinear interaction modeling.This comparison asks whether SQUARE is competitive under fewer trainable parameters, not only against restricted linear adapters.Source: §C.8 · p. 38
  • QAA and QPA test alternative quantum-enhanced PEFT designs, while mechanism-matched classical models cover quadratic, kernel, non-parametric, graph, and manifold approaches.SQUARE without its parameterized circuit is included as an internal ablation of the measured feature map.Source: §C.8 · p. 38
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